Properties

Label 2-55e2-1.1-c1-0-121
Degree $2$
Conductor $3025$
Sign $1$
Analytic cond. $24.1547$
Root an. cond. $4.91474$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·2-s + 2.82·3-s + 3.82·4-s + 6.82·6-s − 2·7-s + 4.41·8-s + 5.00·9-s + 10.8·12-s − 1.17·13-s − 4.82·14-s + 2.99·16-s + 6.82·17-s + 12.0·18-s − 5.65·21-s + 2.82·23-s + 12.4·24-s − 2.82·26-s + 5.65·27-s − 7.65·28-s + 3.65·29-s − 1.58·32-s + 16.4·34-s + 19.1·36-s + 7.65·37-s − 3.31·39-s − 6·41-s − 13.6·42-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.63·3-s + 1.91·4-s + 2.78·6-s − 0.755·7-s + 1.56·8-s + 1.66·9-s + 3.12·12-s − 0.324·13-s − 1.29·14-s + 0.749·16-s + 1.65·17-s + 2.84·18-s − 1.23·21-s + 0.589·23-s + 2.54·24-s − 0.554·26-s + 1.08·27-s − 1.44·28-s + 0.679·29-s − 0.280·32-s + 2.82·34-s + 3.19·36-s + 1.25·37-s − 0.530·39-s − 0.937·41-s − 2.10·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(24.1547\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.674340519\)
\(L(\frac12)\) \(\approx\) \(8.674340519\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 - 2.41T + 2T^{2} \)
3 \( 1 - 2.82T + 3T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 - 1.65T + 59T^{2} \)
61 \( 1 - 9.31T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 1.17T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.543595794065615730625616272981, −7.86162874918331236731681670897, −7.09444665063039379876237783115, −6.42934232509989407182661143997, −5.48530395293167692716477900078, −4.65277066717994482718023454751, −3.77809346775746328276417327705, −3.10348990680387451660398411655, −2.78585550565994987651273464998, −1.58298745393235635248050307252, 1.58298745393235635248050307252, 2.78585550565994987651273464998, 3.10348990680387451660398411655, 3.77809346775746328276417327705, 4.65277066717994482718023454751, 5.48530395293167692716477900078, 6.42934232509989407182661143997, 7.09444665063039379876237783115, 7.86162874918331236731681670897, 8.543595794065615730625616272981

Graph of the $Z$-function along the critical line